This algebraic description should then be complemented by a topological approach,
leading for example to the notion of continuous
functions. This is achieved by completing the algebra
generated by the noncommutative coordinates to a
\(C^\star \) -algebra. Typically \(C^\star \) -algebras arise as algebras of
operators on a Hilbert space. Connes noncommutative geometry [1]} starts from
these notions and enriches the \(C^\star \) -algebra structure and its representation
on a Hilbert space, generalizing to the noncommutative case also the
notions of smooth functions and metric structure.
